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We describe simply connected compact exceptional simple Lie groups in very elementary way. We first construct all simply connected compact exceptional Lie groups G concretely. Next, we find all involutive automorphisms of G, and determine…

We provide explicit generators and relations for the affine Kac-Moody groups, as well as a realization of them as (twisted) loop groups by means of Galois descent considerations.

We show how to use topological ideas, such as compactness, to establish orderability properties of infinite groups. A new application is to provide a left-ordering for the group of PL homeomorphisms of a connected surface with boundary…

We work in a first-order setting where structures are spread out over a metric space, with quantification allowed only over bounded subsets. Assuming a doubling property for the metric space, we define a canonical {\em core} $\mathcal{J}$…

We prove a theorem of Tits type about automorphism groups for compact Kahler manifolds, which has been conjectured in the paper [KOZ].

We consider Tonelli Lagrangians on a graph, define weak KAM solutions, which happen to be the fixed points of the Lax-Oleinik semi-group, and identify their uniqueness set as the Aubry set, giving a representation formula. Our main result…

Using small cancellation for rotating families of groups, we construct new examples of aspherical polyhedra.

This paper is on the inverse parameterized differential Galois problem. We show that surprisingly many groups do not occur as parameterized differential Galois groups over K(x) even when K is algebraically closed. We then combine the method…

We generalise the definition of a group algebra so that it makes sense for non-locally compact topological groups, in particular, we require that the representation theory of the group algebra is isomorphic (in the sense of Gelfand-Raikov)…

We develop the representation theory for reductive linear differential algebraic groups (LDAGs). In particular, we exhibit an explicit sharp upper bound for orders of derivatives in differential representations of reductive LDAGs, extending…

We give a criterion for two l-adic Galois representations of an algebraic number field to be isomorphic when restricted to a decomposition group, in terms of the global representations mod l. This is applied to prove a generalization of a…

We study the model theory of covers of groups definable in o-minimal structures. This includes the case of covers of compact real Lie groups. In particular we study categoricity questions, pointing out some notable differences with the case…

We study generalized Lie bialgebroids over a single point, that is, generalized Lie bialgebras. Lie bialgebras are examples of generalized Lie bialgebras. Moreover, we prove that the last ones can be considered as the infinitesimal…

We create plots of algebraic integers in the complex plane, exploring the effect of sizing the integers according to various arithmetic invariants. We focus on Galois theoretic invariants, in particular creating plots which emphasize…

This thesis studies arithmetic of linear algebraic groups. It involves studying the properties of linear algebraic groups defined over global fields, local fields and finite fields, or more generally the study of the linear algebraic groups…

Some results that are true in classical groups are investigated in generalized groups and are shown to be either generally true in generalized groups or true in some special types of generalized groups. Also, it is shown that a Bol groupoid…

This should be the final version of this paper. Numberous minor improvements have been made to the manuscript, one argument has been corrected, and an appendix has been added.

Locally conformal symplectic (l.c.s.) groupoids are introduced as a generalization of symplectic groupoids. We obtain some examples and we prove that l.c.s. groupoids are examples of Jacobi groupoids in the sense of \cite{IM}. Finally, we…

Topos properties of the category of covering groupoids over a fixed groupoid are discussed. A classification result for connected covering groupoids over a fixed groupoid analogous to the fundamental theorem of Galois theory is given.

We introduce a new graph invariant of finite groups that provides a complete characterization of the splitting types of unramified prime ideals in normal number field extensions entirely in terms of the Galois group. In particular, each…